Predictive Mobility Model for β-Ga₂O₃ at Cryogenic Temperature

Abstract

In this work, the transport properties of charge carriers in $\beta$-Ga${}_2$O${}_3$ were investigated, along with intrinsic physical mechanisms such as lattice vibrations, impurity scattering, and interfacial effects. The high-field behavior of carrier mobility was characterized using vacuum deposition techniques for the fabrication of electrodes with ohmic contacts, and the Hall effect measurement system was employed to test the temperature-dependent mobility of Sn-doped n-type (100) and (001) $\beta$-Ga${}_2$O${}_3$ samples at a cryogenic temperature. A predictive model for $\beta$-Ga${}_2$O${}_3$ mobility was developed by examining the effects of the temperature on the scattering mechanisms based on a theoretical transport model. The experimental results for $\beta$-Ga${}_2$O${}_3$ mobility, which varied with the temperature and doping concentration, showed good agreement with the theoretical model within the temperature range of 15–300 K. The maximum discrepancy between the predictive model and the experimental data was less than 5%. This study provides valuable theoretical insights for the design and simulation of $\beta$-Ga${}_2$O${}_3$ devices.

1. Introduction

As a representative example of emerging semiconductor materials, gallium oxide ($\beta$-Ga${}_2$O${}_3$) exhibits exceptional properties with a wide bandgap of 4.9 eV, a high breakdown field strength of 8 MV/cm, and a Baliga’s Figure of Merit reaching up to 3444 MW/cm${}^2$ [1,2,3]. The listed research results show that $\beta$-Ga${}_2$O${}_3$ demonstrates significant potential for use in high-voltage and high-power electronic devices [4,5,6]. Compared to materials such as silicon (Si) and gallium nitride (GaN), which have been extensively researched, research on $\beta$-Ga${}_2$O${}_3$ is currently still focused on the development of single large-sized crystals and the optimization of device design [7,8,9]. Although preliminary results have been obtained for $\beta$-Ga${}_2$O${}_3$ in experimental research and application development, in-depth research on the electrical properties of this material, especially regarding mobility models, is not systematic, which limits its application in high-performance electronic devices [10].

In this paper, the transport characteristics of the carriers in $\beta$-Ga${}_2$O${}_3$ and the various scattering mechanisms affecting the carriers were studied. By integrating the impacts of different doping levels and crystal orientations, a $\beta$-Ga${}_2$O${}_3$ mobility model was established. Additionally, a series of variable-temperature mobility tests were conducted on Sn-doped n-type (100) and (001) $\beta$-Ga${}_2$O${}_3$ using a Hall effect testing system combined with a vacuum deposition technique, covering a temperature range from 15 to 300 K. The exhaustive analysis and fitting of the experimental data yielded models of $\beta$-Ga${}_2$O${}_3$ mobility that characterize different crystal orientations and doping levels.

2. Methodology

2.1. Experimental Details

The mobility characterization experiment was based on the theory of the Hall effect, utilizing the Van der Pauw method and the Hall effect testing system provided by the Lake Shore Corporation. This study involved the assessment of the mobility of single Sn-doped (doping concentration of ≥1 ×10${}^{18}$ cm${}^{-3}$) $\beta$-Ga${}_2$O${}_3$ crystals with (001) and (100) orientations and dimensions of 10.0 mm × 10.5 mm within a temperature range from 15 to 300 K. Titanium (Ti) with a thickness of 1500 nm and dimensions of 1.0 mm × 1.0 mm was deposited at the corners of the $\beta$-Ga${}_2$O${}_3$ square using the vacuum evaporation coating method to form ohmic contacts, as shown in Figure 1a. The production of a protective metal layer followed to prevent Ti oxidation. Post-deposition, the samples were annealed at 450 ${}^{\circ }$C to minimize the Ti-$\beta$-Ga${}_2$O${}_3$ interface defects, ensuring low resistance and optimal conductivity for testing. After confirming good contact, a molecular pump was used to evacuate the chamber, ensuring that the vacuum level was reduced to below 5.0 × 10${}^{-4}$ mbar, at which point the temperature-dependent experiment could begin. The resistivity test was conducted by applying a constant excitation current of 10 mA, while the Hall voltage test was carried out under an excitation magnetic field of 0.7 T. The test was performed continuously during the cooling process to ensure that the electrical property changes of the sample could be recorded in real time as the temperature changed, thereby obtaining detailed temperature-dependent data.

2.2. Theoretical Model

The Caughy–Thomas equation delineates the interdependence between the carrier concentration and mobility, allowing for the derivation of key parameters to model mobility as a function of the carrier concentration in $\beta$-Ga${}_2$O${}_3$. The model integrates impurity and lattice scattering effects, derived from the analysis of a low-field carrier concentration dependence [11,12,13]. At 300 K, it represents semiconductors like Si, SiC, and GaN well. The formula captures the carrier concentration–mobility relationship:

$${\mu }_N={\mu }_{min}+{\displaystyle \frac{{\mu }_{max}-{\mu }_{min}}{1+{\left(\frac{N}{N_{\mathrm{ref}}}\right)}^{\gamma }}}.$$

(1)

Here, $\mu$${}_{min}$ and $\mu$${}_{max}$ denote the minimum and maximum mobility observed within the experimental concentration range; N${}_{ref}$ signifies the reference concentration; and the parameter $\gamma$ is designated as a fitting parameter within the model.

The temperature-dependent mobility of carriers across various doping levels was scrutinized, revealing that phonon scattering predominantly affects mobility at low doping levels and room temperature. Other scattering mechanisms were considered negligible, with the maximum carrier mobility assumed to occur under lattice scattering conditions:

$${\mu }_{\max }={\mu }_L.$$

(2)

Further exploration into the effect of the temperature on phonon scattering elucidated that, in accordance with Equation (2), lattice vibrations undergo an exponential alteration as the temperature rises. The exponential factor $\theta$${}_1$ was introduced to gauge the impact of temperature fluctuations on the maximum carrier mobility, expressed as $\mu$${}_{max}$:

$${\mu }_{max}={\mu }_{max}\left(300\mathrm{K}\right){\left({\displaystyle \frac{T}{300\mathrm{K}}}\right)}^{{\theta }_1}.$$

(3)

The equation illustrates that lattice scattering’s influence increases with the temperature. Temperature variations also impact the ionized impurity scattering efficiency, notably in heavily doped cases. Thus, the temperature’s effect on the carrier concentration must be considered. Integrating the temperature and concentration dependencies of mobility, a comprehensive model for $\beta$-Ga${}_2$O${}_3$ mobility was developed with the fitting parameters $\theta$${}_1$ and $\theta$${}_2$:

$$\mu ={\mu }_{min}+{\displaystyle \frac{{\mu }_{max}\left(300\mathrm{K}\right){\left(\frac{T}{300\mathrm{K}}\right)}^{{\theta }_1}-{\mu }_{min}}{1+{\left(\frac{N}{N_{ref}\left(300\mathrm{K}\right)}\right)}^{\gamma }·{\left(\frac{\mathrm{T}}{300\mathrm{K}}\right)}^{{\theta }_2}}}.$$

(4)

3. Predictive Model

3.1. Effects of Doping Concentration on Mobility

When considering the (001) and (100) crystal planes separately, the scarcity of available experimental data necessitated the development of predictive models to analyze the mobility trends as a function of the temperature and concentration. After obtaining an adequate data set, we intended to refine these models. The variations in the carrier mobility in relation to the doping concentration for (100) $\beta$-Ga${}_2$O${}_3$ and (001) $\beta$-Ga${}_2$O${}_3$ were fitted individually, with the corresponding fit curves and experimental data depicted in Figure 2. Specifically, Figure 2b illustrates the fitting outcomes for (100) $\beta$-Ga${}_2$O${}_3$, which had a precision of 99%, while Figure 2a displays the fitting results for (001) $\beta$-Ga${}_2$O${}_3$, with a fitting error of below 5%.

Table 1. Fitting parameters.

Crystal Planes	${\mathit{\mu }}_{\mathbf{max}}$	${\mathit{\mu }}_{\mathbf{min}}$	${\mathit{N}}_{\mathbf{ref}}$	$\mathit{\gamma }$	${\mathit{\theta }}_1$	${\mathit{\theta }}_2$
	[cm${}^{\mathbf{2}}$ V${}^{-\mathbf{1}}$s${}^{-\mathbf{1}}$]	[cm${}^{\mathbf{2}}$ V${}^{-\mathbf{1}}$s${}^{-\mathbf{1}}$]	[cm${}^{-\mathbf{3}}$]
(100)	64.7	46.5	5.48 × 1018	6.14	0.49	0.29
(001)	82.9	39.1	5.22 × 1018	7.50	0.86	0.40

provides a detailed description of the parameters derived from the fitting process.

By incorporating the fitting parameters into Equation (1), the physical models that depict the mobility variation of Sn-doped gallium oxide with the impurity concentration for the (100) and (001) planes at 300 K are presented as follows:

$${\mu }_{\left(100\right)}=46.5+{\displaystyle \frac{18.2}{1+{\left(\frac{5.48\times {10}^{18}}{N}\right)}^{6.14}}}.$$

(5)

$${\mu }_{\left(001\right)}=39.1+{\displaystyle \frac{43.9}{1+{\left(\frac{N}{5.22\times {10}^{18}}\right)}^{7.50}}}.$$

(6)

3.2. Unified Mobility Model

Based on the parameter fitting values shown in Table 1, in conjunction with the variable-temperature mobility test results depicted in the figures, the formula was fitted and verified, and the corresponding parameters within the formula were extracted. The fitting results are shown in Figure 3. The overall trends in the carrier mobility from a low temperature to room temperature were nearly identical, although there were numerical differences. The (100) orientation of $\beta$-Ga${}_2$O${}_3$ exhibited a wider mobility range than the (001) orientation during variable-temperature tests due to its denser atomic arrangement, increasing the carriers’ susceptibility to lattice scattering [19]. Temperature changes altered the lattice vibrations, causing greater mobility fluctuations in the (100) samples. Both orientations showed a similar temperature–mobility pattern, with carriers reaching excitation saturation at above 200 to 300 K. As the temperature increased, the carrier concentration remained largely constant, but the heightened temperature accelerated electron movement, thus augmenting the likelihood of collisions. The dominant factor affecting mobility was lattice scattering, which led to a downward trend. Impurity ionization was the main source of carriers as the temperature rose from a low temperature to 200 K, at which there was a lower carrier concentration. In this scenario, the temperature was the primary factor influencing the carrier movement velocity; the higher the temperature, the more vigorous the thermal motion of the carriers, leading to increased mobility.

The theoretical analysis model and experimental results for the $\beta$-Ga${}_2$O${}_3$ samples had a fit error of less than 5%. The experimental data for $\beta$-Ga${}_2$O${}_3$ with a (001) orientation aligned well with theoretical models in a high-temperature regime, whereas discrepancies emerged between the experimental observations and theoretical predictions at temperatures below 150 K. This deviation was attributed to the pronounced influence of the intrinsic defects within the gallium oxide crystal lattice on the carrier mobility under low temperatures. The fitting parameters are listed in Table 1. By substituting these parameters into Equation (4), a physical model that characterized the mobility of (001) and (100) $\beta$-Ga${}_2$O${}_3$ as a function of the concentration and temperature variation could be obtained, as shown in Equations (7) and (8).

$${\mu }_{\left(100\right)}=46.5+{\displaystyle \frac{64.7\times {\left(\frac{T}{300\mathrm{K}}\right)}^{0.49}-46.5}{1+{\left(\frac{5.48\times {10}^{-18}}{N}\right)}^{6.14}\times {\left(\frac{T}{300\mathrm{K}}\right)}^{0.29}}}.$$

(7)

$${\mu }_{\left(001\right)}=39.1+{\displaystyle \frac{82.9\times {\left(\frac{T}{300\mathrm{K}}\right)}^{0.86}-39.1}{1+{\left(\frac{N}{5.22\times {10}^{18}}\right)}^{7.50}·{\left(\frac{T}{300\mathrm{K}}\right)}^{0.40}}}.$$

(8)

4. Conclusions

A comprehensive $\beta$-Ga${}_2$O${}_3$ mobility test was successfully conducted across an exceptionally broad temperature spectrum. A robust model of gallium oxide mobility was developed after the empirical data were meticulously aligned with a theoretical model of mobility. The relationship between the carrier mobility and temperature is encapsulated in this model, which provides a predictive framework for understanding the behavior of gallium oxide under various thermal conditions.